Three Dimensions in Motivic Quantum Gravity

M. D. Sheppeard December 2018

Abstract Important polytope sequences, like the associahedra and permutohedra, contain one object in each dimension. The more particles labelling the leaves of a tree, the higher the dimension required to compute physical amplitudes, where we speak of an abstract categorical dimension. Yet for most purposes, we care only about low dimensional arrows, particularly associators and braiding arrows. In the emerging theory of motivic quantum gravity, the structure of three dimensions explains why we perceive three dimensions. The Leech lattice is a simple consequence of quantum mechanics. Higher dimensional data, like the e8 lattice, is encoded in three dimensions. Here we give a very elementary overview of key data from an axiomatic perspective, focusing on the permutoassociahedra of Kapranov. These notes originated in a series of lectures I gave at Quantum Gravity Re- search in February 2018. There I met Michael Rios, whom I originally taught about the associahedra and other operads many, many years ago. He discussed the magic star and exceptional periodicity. In March I met Tony Smith, the founder of e8 approaches to quantum gravity. In June I attended the Advances in Quantum Gravity workshop at UCLA, where I met the M theorist Alessio Marrani, and in July participated in Group32 in Prague, where I met Piero Truini, who I later talked to about operad polytopes in motivic quantum gravity. All these activities led to a greatly improved understanding of the emerging theory. I was then forced to return to a state of severe poverty, isolation and abuse, ostracised eﬀectively from the professional community, from local com- munities, from family, and many online websites. I am working now on an ipad in my tent. Although I hope the discussion here is elementary, the motivation for it comes from deep mathematical conundrums, as any good question in quantum gravity should. As a category theorist, the slow development of topological ﬁeld theories was endlessly frustrating, with its separation of spatial and algebraic data. Like in gauge theories, one was supposed to mark spatial diagrams with objects from an algebraic category, such as a traditional category of Hilbert spaces. But the most interesting functors are monadic, and it seemed that gravity should canonically interpret its geometry in an algebraic way. Moreover, the common habit of grabbing R or C as a God given set of numbers was in

1 conﬂict with both the axioms of a topos [1] and the need to construct classical manifolds from discrete quantum data. We begin with the power set monad [1], which simply maps any set to the set of all subsets of the set. In the case of a three element set, there are eight possible subsets including the empty set, listed by the vertices of ﬁgure 1. A set with n elements gives a cube in dimension n. An edge on a cube, directed away from the empty set 1, is an inclusion of sets. The concatenation of elements, like αiαj, denotes the union. Conversely, reverse arrows restrict to a subset of a set, and the source of a square is the intersection of sets. A category theorist talks about the duality of reversing arrows, or the opposite category of the category of all sets. But quantum mechanics is not about sets. Axioms based on set theoretic mathematics must be replaced by axioms for quantum logic. In the next section we see that n-cubes also play an important role in the quantum case. The signed vertices on a cube denote a state of n qubits in the theory of quantum computation [2][3][4], and we use the universality of special categories like the Fibonacci anyons [5][6] to argue that gauge groups emerge from braid group representations.

Figure 1: The parity cube in dimension 3

From cubes we move on to the associahedra [7] and permutohedra [8], the polytopes that are specified by rooted binary trees or permutations. The combi- nation of these polytopes, including both Mac Lane pentagons [9] and braiding hexagons, are the permutoassociahedra of [10]. In particular, the 120 vertex permutoassociahedron in dimension 3 stands in for half the 240 roots of the 8 dimensional e8 lattice. So what is a motive? Certainly, it does not begin with a variety based on C, or any other random field. A whole, infinite dimensional category of motives

2 should be linked to our quantum axioms, taking motivic diagrams directly to physical results, following the original vision of Grothendieck [11]. A classical spacetime is a mere afterthought, a three time Minkowski space perhaps com- pactiﬁed to SU(2) × SU(2), constructed using the anyon braid group B3 [12]. Ribbon diagrams are spatial, and integral octonions, for instance, are algebraic, but the category of motives thinks they are one and the same, thinks that a fundamental group drawn in knots is its own algebraic structure. And the Standard Model of particle physics lives here.

1 A motivic perspective

Geometric categorical axioms encode algebraic information. In the theory of quantum gravity, we require a canonical symbiosis of discrete geometry and algebraic rules. Motivated by the power set monad [1], we begin with the humble cube. The vertices of figure 1 carry two sets of labels. Physically, the signs encode the fractional charges of the anyons [13] underlying the leptons and quarks, so that −−− denotes 3 zero charges on a neutrino and +++ denotes the charge on a positron. There is a second cube for the negative leptons and quarks [14][15]. In quantum computation, three signs stand for a set of 3 qubit states. At the same time, the edges of a 3-cube move in the 3 directions of a qutrit space. A qutrit has three possible measurement outcomes, while a qubit has two. A path from the source 1 to the target α1α2α3 has three edges. These six paths clearly form a copy of the permutation group S3, if we label the directions 1, 2 and 3. Seven of the vertices give an unconventional Fano plane, as shown in figure 2. This gives us a basis for the integral octonions [16][17], which are appearing here automatically, once we decide to study sets carefully. In quantum mechanics, an element of a set usually becomes a basis vector in a state space. Instead of three points and their unions, consider three generic lines in a Euclidean plane with respect to intersection in the plane. If no lines are selected, we have the whole plane. If one line is selected, we get a line. If two lines are selected, we obtain the point of intersection of the lines. And finally, if all three lines are selected, we have the empty set. Observe how these possibilities reverse the position of empty set, moving it from 1 to the target of the cube. The source 1 is now the entire plane, an axiomatic object of dimension 2. The singleton αi are now dimension 1 objects and the αiαj of dimension zero. Whereas sets are always zero dimensional, whatever the cardinality, quantum geometry interprets a whole number n as a dimension. In this example, there are spaces of dimension 0, 1 and 2. A classical topolo- gist would take these numbers and put them on the vertices of a triangle, calling the triangle a 2-simplex. We have already drawn the 2-simplex as the interior of three lines in a plane. It is also the diagonal slice of the 3-cube at the αi positions, or rather, the points 001, 010 and 100. Now consider qutrit words which allow more than one step in each direction. As well as 231 ∈ S3 we have 2231, taking two steps in one direction. Allowing arbitrary noncommutative words,

3 Figure 2: The Fano plane we see discrete paths on rectangular blocks in any dimension. On a cubic array of paths, the diagonal slice ﬁxes the length of the word, so that length 3 qutrit words collapse to the discrete simplex of ﬁgure 3. Simplices carry commutative words, while cubes carry the full set of noncommutative words. Given any noncommutative word, a rectangular subspace of subwords sits below it. For example, 2231 sits on a 3-cube with three points along each edge, while its subpath 231 lies in the space of paths closer to the source vertex. 2 If the stepping represents powers of a set of primes, as in p2p3p1, then the subspace vertices list the divisors, and each dimension out to ∞ lists powers of an integral prime. Then the positive integers in Z use up every cubical point in every dimension. A qubit corresponds to the prime 2 and a qutrit to the prime 3, because state spaces are characterised by Schwinger’s mutually unbiased bases [18][19][20]. For qubits, the three Pauli matrices are replaced by the three unitary bases of eigenvectors,

1 1 1 1 1 i 1 0 F2 = √ ,R2 = √ ,I2 = . (1) 2 1 −1 2 i 1 0 1

4 Figure 3: The commutative tetractys for three qutrits

For qutrits, the bases are

1 1 1 1 ω 1 1 1 ω 1 1 −1 1 F3 = √ 1 ω ω ,R3 = √ 1 1 ω ,R3 = √ ω 1 1 ,I3, 3 1 ω ω 3 ω 1 1 3 1 ω 1 (2) where ω is the cubed root of unity. In every prime power dimension d, there is a (one step) circulant generator Rd whose powers give d bases that are mutually unbiased. Thus each cubic point is associated to a set of d = pr matrices. Dimension 3 is special because circulants are Hermitian, like the elements of the exceptional Jordan algebra over the octonions [16][17]. The physical charges attached to figure 1 by the C ⊗ O algebra [14][15] correspond to ribbon twists in the particle spectrum of Bilson-Thompson [13][21], where three anyon strands braid in B3 to give leptons and quarks. Sometimes we view the αi as differential forms, but we want to avoid any ref- erence to manifolds over bothersome fields. Most√ complex numbers are obtained from the (half)√ integers in dimension√ 8 using Q( 5) with the golden ratio. Let φ = (1 + 5)/2 and ρ = φ + 2. Then the Gaussian integer combinations a + bφ + cρ + dφρ fill the plane. Everything begins with discrete data.

5 2 Associahedra and Permutohedra

The permutation group Sn has a polytope in dimension n − 1. The three dimensional permutohedron is shown in ﬁgure 4 with its S4 labels. Think of the

Figure 4: Three dimensional permutohedron elements of S4 as integral coordinates in dimension 4 and restrict to the three dimensional plane given by the fixed sum 1 + 2 + 3 + 4 = 10. The infinite set ` Sn of all permutohedra form an operad, meaning that natural compositions of permutations give larger permutations. Another important operad is the set of associahedra polytopes [7], whose vertices are labeled by binary rooted planar trees. Figure 5 gives the pentagon associahedron for trees with 4 leaves. As an axiom for categories of Hilbert spaces (symmetric monoidal) the pentagon edges are directed associator maps (12)3 → 1(23) between bracketing options on nonassociative words. In every dimension n, the associahedron is embedded in a discrete simplex by the coordinates of [8]. The pentagon lies in the bottom left corner of the tetractys of figure 3, as follows. Take vertices such that each place i in the word has a value that is less than or equal to i. This is the set 111, 112, 122, 113 and 123. These words may be viewed (homework problem) geometrically as arbitrary forests of rooted trees on three nodes. Forests are geometrically dual to binary rooted trees. For the 14 vertex associahedron we use length 4 words in 4 letters, but the 14 points required actually fit inside a three 4-dit simplex, which itself sits

6 Figure 5: Mac Lane’s pentagon axiom inside the three dimensional path cube with four points along each edge. This 64 point cube is the true home of the tetractys. The edges of an associahedron are marked by a tree with one ternary node, which collapses the binary nodes on the end vertices. Similarly, each face of an associahedron is a tree with two collapsed nodes, and each cell in any dimension is labeled by a tree. The associahedra operad underlies scattering amplitude calculations in particle physics [22][23]. Particle number determines the required dimension of the calculation, which is therefore arbitrary. There are only two ways to choose two prime powers that diﬀer by 1. We can either put 2 with 3 or 3 with 4. The latter gives our 64 point cube with internal three qutrit tetractys, and the former a simple cube with 8 vertices. Including both noncommutative qubits and qutrits, these state space dimensions are 1728 and 24. The special number 1728 is the natural normalisation for the j-invariant. Observe that labels on the leaves of a tree on the pentagon give an element of S4. Allowing all possible labels on all trees, we obtain the three dimensional permutoassociahedron [10] of ﬁgure 7. We suspect now that cubes, simplices, associahedra and permutohedra are all closely related. In fact, they carry a rich algebraic structure, crucial to the observational arguments of the neutrino CMB correspondence [12][24]. We discuss now Solomon’s algebra of descent, which maps the permutohedra to the sign labels on a cube of the same dimension.

7 3 Cubes and Solomon’s Algebra

Finite sets are acted on by permutations. Taking an element of the hexagon S3, we permute the αi vertices on the cube, or reorder the word α1α2α3. Distinct actions slice a cube into its diagonal subsets. The cardinality of a diagonal is a binomial coefficient, and these sum to the total 2n. There are 12 ways to braid three objects into a permutation in S3, either by flipping only two objects or by flipping one object around the other pair. These are the 12 edges of the octahedron in figure 6. The bracket on a word indicates

Figure 6: Octahedron for permutations the face on the cube dual to the octahedron, where a permutation is a three step path along edges on the cube. Accounting for all possible nonassociative words in S3, there are two copies of the octahedron, giving all vertices of the 12-gon permutoassociahedron [10], using both swap maps and associators between the two octahedra. That is, we use three vertices in one octahedron, three in the other, and alternate these with the six associators in between. Why such an abtuse drawing of a 12-gon? Because what we aim for, in the next section, is a special set of three dimensional pictures that capture the data we need to describe gravity. The 12-gon carries all possible terms in weak

8 Jacobi rules, such as those required for the nonassociative algebras of exceptional periodicity based on the magic star [25][26]. Return to the permutohedron of ﬁgure 4. This polytope collapses to a 3-cube with signed vertices. The signs at each vertex of Sn are the signature of the permutation, given by following the increases and decreases in the numerical entries from left to right. Signs for S3 appear in ﬁgure 6. For example, the signature of 4312 is − − +. Signature sets occupy neighbouring sets in the Sn polytope. Solomon’s (Hopf) algebra [27] on signature classes uses integral elements of the group algebra product for Sn. An element of this group algebra, over Z, is an arbitrary sum of permutations, written out like a polynomial. The product is like an ordered polynomial product. For example,

(231 + 312)(231) = 231 · 231 + 312 · 231 = (312 + 123). (3)

For each signature of Sn write the sum of permutations with that signature. For example, +− = (132 + 231). (4) Solomon’s theorem shows that the polynomial product on the signatures always results in a signature, closing an algebra on the signed cube.

Figure 7: The permutoassociahedron

9 The permutoassociahedra vertices in dimension n list all associahedra trees with permutation labels, characterising the essential data for the scattering of n + 1 particles. However, the essential axioms belong to categories in three dimensions: the braided monoidal categories of quantum computation. We are interested in finding the three dimensional essence of these polytopes, for application to quantum gravity. This means defining a sequence of trivalent graphs, picking out three special neighbouring permutations to any permutation in Sn. In the case of S3, there are three other permutations resulting from a flip S2. Restricting to a subset of 3 particles in a set of n + 1, we can identify three outgoing edges.

4 The Leech Lattice and Monster

A double copy of figure 7 has 240 vertices, the number of roots in the e8 Lie algebra lattice. The connection between octonions and this lattice was often used in Lie group compactification approaches to gravity, originating decades ago in the work of Tony Smith [28]. To characterise rest mass, we study the relations between 24 dimensional lattices, pertaining to the state space of three qubits and one qutrit. To obtain the 196560 vertices of the Leech lattice Λ24 using three copies of O [29], multiply the 240 by 3 times 273, where 273 = 1 + 16 + 256. That is, double a 3-cube to 16 vertices (particles and antiparticles), double a 7-cube to 256 vertices (magnetic information), and add half a 1-cube. This all makes sense in the Jordan algebra description of quantum gravity, which extends M theory. From moonshine mathematics, we remember that the j-invariant Fourier transform (for three dimensional topological gravity [30]) is closely related to the modular form for the Leech lattice [31][32]. An even nicer polytope decomposition works as follows. As well as the Leech lattice and its automorphism group, we want a description of the Monster M, the largest sporadic finite group [33]. The three largest primes in the order of M give the relation 196883 = (48 − 1)(60 − 1)(72 − 1) (5) for the first non trivial module of M. For us, the fundamental Fibonacci numbers work on a clock of either 12 or 24 ticks, because the Fibonacci sequence mod k for k ≤ 12 has a common cycle length of 240 and factors of k = 12 give a cycle length of 24. A cycle of length 12 appears for mod 8. The importance of the number 24 is why we have 24 hours in a day. Remember, in quantum mechanics every cardinal is a dimension! So hiding in the 196883 are 120 copies of something, which we find in an old paper by Tits [34]. There are 120 elements in the binary icosahedral group, given by the following 4-vectors with quaternion basis {1, i, j, k}. First, 24 vectors coming from the sign choices on (±1, 0, 0, 0) and (±1, ±1, ±1, ±1), with permutations, and also 12 even permutations of each of the 8 sign choices for (0, ±1, ±φ−1, ±φ)/2. The automorphism group of the binary icosahedral group is S5, the 120 vertex

10 Table 1: 1638 vector components total factor v2 576 24 (1, φ2, ρ2) 192 8 (φ2, φ2, 2) 192 8 (φ2, φ2, 2φ2) 192 8 (1, 1, 2) 192 8 (1, 1, 2φ2) 96 4 (1, φ2, φ−2) 96 4 (1, φ2, φ4) 48 2 (0, 2, 2φ2) 24 1 (0, 2, 2) 24 1 (0, 2φ2, 2φ2) 3 0 (0, 0, 4) 3 0 (0, 0, 4φ2) permutohedron on 5 objects. Over Q we have the eight dimensional space of the e8 lattice. How do the golden ratio entries in these vectors mix with entries in a 3- vector? Since any power of φ is linear in φ, the ﬁnal cancellation of φ in the norms requires only that contributing signs sum to zero. We decompose the short vectors of Λ24 as 196560 = 120 · 14 · 117, (6) where each 14 stands for an associahedron in dimension 3. The 14 vertex blowup of S5 is the four dimensional permutoassociahedron, with 1680 vertices (bringing in the seven days). The 14 · 117 copies of 120 are given by the three dimensional vectors of [34], as listed in Table 1. The 14 vertex associahedron has a neat set of hyperplane coordinates in dimension 6. These coordinates count the triangles at a vertex on a hexagon in the chorded hexagon diagrams for binary tree vertices. Ignoring normalisations, we have the points 1. (3, 1, 2, 3, 1, 2) and cyclic permutations 2. (1, 2, 2, 2, 1, 4) and cyclic permutations 3. (1, 3, 1, 3, 1, 3) and (3, 1, 3, 1, 3, 1) away from the squares.

The ﬁrst three coordinates determine the hyperplane, and we are free to subtract 1 everywhere in order to obtain vectors like (2, 0, 1) and (3, 0, 1). Note that φ2 + φ−2 = 3. Why the 117? In future work we will study the connection to quantum logic, especially the Kochen-Specker theorem [35][36] for R3. This canonical proof for the failure of classical ontology uses a conﬁguration of 57 qutrit rays in R3, or 2 rather, points in RP . This gives 114 points on the subspace lattice, to which one usually adds the full plane and the zero space.

11 It seems the entire Leech lattice comes down to the basics of quantum mechanics: the four dimensional permutoassociahedron with 1680 vertices and the logic of Kochen-Specker conﬁgurations.

5 Three Dimensions in Quantum Gravity

The number of vertices in Sn is n!. Starting with the octahedron for S3, we might blow up each vertex by a factor of 4 to find the 24 vertices of figure 4. Indeed, the truncated octahedron is the permutohedron polytope. For each σ ∈ S3 there are 4 positions to place the letter 4 to make an element of S4. For how long can we keep going this way? Blowing up each vertex of S4 by a pentagon gives the permutoassociahedron, now a representation of S5. Many higher dimensional edges are lost as blowups continue in dimension 3, but the set Sn is recovered. Let us look more closely now at the trivalent nature of restricted Sn polytopes. For each blowup by an n-gon, the n edges that are not used in drawing the n-gon must join a neighbouring n-gon vertex. So we must partition these n edges into three sets, to follow the original three edges of the initial graph. For n mod 3 ≡ 0, an equal partition is possible and the new graph is clearly well defined. For n mod 3 ∈ 1, 2, one out of three outgoing edges is the odd man out. How do we know we can define a new polytope with the odd man edges connecting vertices consistently? If there was a Hamiltonian circuit on the initial Sn polytope, then each trivalent vertex automatically has an odd man out missing from the circuit. Observe that S4 and figure 7 both have Hamiltonian circuits. Since S4 has a Hamiltonian circuit, and 5 ≡ 2 is an odd man case, the pentagon blowup exists as figure 7 (ignoring for now the 4-valent part). Does a Hamiltonian circuit always exist on the new graph? In general, finding circuits is an NP -complete problem, but we have a specific sequence of graphs. Let us first build a set of loops in the blowup using the original circuit as a template. The set covers every vertex once. Since an edge set from one polygon to another creates a bunch of square faces, one can only cover all vertices on an n-gon if one goes in and/or out across a leg using neighbouring edges. We stick to two pairs of edges. On each edge bunch, use either the first two or last two edges. These pairs can always link edges around the polygon in an obvious way, so that every vertex is covered. Now there are two circuits around the original graph, one for each of the two edges along a bunch. For a Hamiltonian circuit, we need to join these two. This is easily done by flipping the edge pair around one single square, since the used two edges on the bunch belong one to each circuit. We have established the following. There exists a sequence of trivalent polytope graphs with Hamiltonian circuits on n! vertices for any n ≥ 3, each graph obtained from the previous one by an n-gon blowup at each vertex. Figure 8 paints a Hamiltonian circuit on S5. Its blowup is S6, with 720

12 Figure 8: A Hamiltonian circuit on the permutoassociahedron vertices. From these three dimensional reduced permutohedra, we obtain a set of reduced associahedra. Each σ ∈ Sn is represented by a binary tree whose nodes have distinct levels, as opposed to the trees of figure 5, where the word (12)(34) does not distinguish the order of the bracketing. The hexagon S3 collapses to the pentagon. The S4 polytope collapses to the three dimensional associahedron with the contraction of three squares joined to two hexagons. Define the reduced associahedron for Sn in dimension 3 by collecting vertices according to the tree type labels. Finally, an associahedron permutohedron pair, each from dimension 3, defines a low dimensional reduced permutoassociahedron [10]. In three dimensions, we play with 3×3 matrices. The 27 dimensional exceptional Jordan algebra is defined using three qutrits, following [37]. These 1, 2, 3 valued qutrits are the powers i, j and k of the matrices

ωiXjP k, (7) for X and P given by

0 1 0 1 0 0 X = 0 0 1 ,P = 0 ω 0 . (8) 1 0 0 0 0 ω

Letters X and P stand for discrete space and momentum. Recall that the 27 words label the cubic paths for the tetractys of ﬁgure 3. This Jordan algebra J of Hermitian matrices over the octonions marks a vertex on the magic star [25] characterisation of the algebra e8 [26]. There is a triangle of J algebras and a triangle of J. We can obtain the 13 points of a magic star from a divided cube

13 by projection along the diagonal. The source, target and centre point go to the centre of the star.

6 Comments

With almost no input, we have uncovered some of the algebraic ingredients of quantum gravity. As is well known, physical rest masses for leptons and quarks are grouped as eigenvalues of 3 × 3 Hermitian matrices with special phases related to charge [12]. The 3-cube charges augment the basis neutrino state. The essential Koide-Brannen phase for leptons is 2/9, which is the ratio of six paths on a cube to the 27 paths of the tetractys, sitting inside the larger cube for Dirac spinors. The foundational neutrino phase is of course π/12. The duality between the CMB neutrino cutoff (maximal cosmological wave- lengths) and the Planck scale defines a Higgs scale via the inverse see-saw √ mH = mν mPl [12], and we see quantum masses growing from the pairing of (right handed) CMB neutrinos with local neutrino states. Everything is governed by the laws of computation, when axiomatised in the infinite dimensional setting of quantum logic. From state sum triangulations, we know that the dynamics of four dimensions requires a braiding, as shown in figure 9, where one pentagon chord is forced to overlap another. Without this single edge, the 4-simplex is a double tetrahedron in three dimensions. A 4-

Figure 9: 3d simplex and 4d simplex simplex then becomes a planar pentagon, a face of ﬁgure 7. Instead of a 7-sphere for the roots of e8, we look at the doubled permutoassociahedron. By reducing dimension using categorical information, the algebra of gravity manifests itself in an abstract braid soup (a la Langlands).

14 References

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